This post is a collaborative product of the MathTwitterBlogosphere

During our second trimester this school year some of our instructional goals centered around integers and integer arithmetic.

6.NS.C.5 – Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of 0 in each situation.

6.NS.C.6 – Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane.

7.NS.A.1 – Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.

We began our investigation of integers using a sequence of open number sentences from Project Z. This came as a result of posts from Michael Pershan and Kent Haines. Students were able to make sense of and discuss the actions taking place during integer arithmetic.

The green and red arrows should be pointing to the same space on the line. My bad!

Then after some twitter collaboration with Michael and Kent we continued to investigate open number sentences.

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Its week two of the 2016 MTBoS Blogging Initiative, so here goes…

The prompt for this week’s blog post is as follows:

Our week two blogging challenge is to simply blog about one of your favorite things.  Called a “My Favorite,” it can be something that makes teaching a specific math topic work really well.  It does not have to be a lesson, but can be anything in teaching that you love!  It can also be something that you have blogged or tweeted about before.  Some ideas of favorites that have been shared are:

• A lesson (or part of one) that went great
• A game your students love to play
• A fun and/or effective way to practice facts
• A website or app you love to use in class
• An organizational trick or tip that has been life changing
• A product that you use in your classroom that you can’t live without!

My favorite is none of these things, but indirectly related to all of them.  My favorite thing in teaching is my students, all of them! Even the ones whose regular review of my class sounds little something like this, “Ugh, Mister, that’s mad work!” All in a day’s work, friends, all in a day’s work!

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What follows is my submission to the MathTwitterBlogoSphere 2016 Blogging Initiative…

The MTBoS is a group of passionate mathematics educators who take to twitter and blogs to share what it means to be a math educator. This could be great lessons, reflection on one’s day, or a question relating to one’s practice. If you are looking for a PLC, there is one just waiting for you on the interweb!

I began this post trying to reflect about ONE good thing that happened during my day, but invariably when you begin thinking about good things more and more come to light. Now this post has morphed into the one good thing that happened during each period today. All of my classes are studying measurement and as a special education teacher each class is focused on a different aspect of measurement. I had 3 classes today (one class was on a trip and missed my period with them). Here is what the focus of each of our classes was…

• Class 1 – Using a ruler to model placing fractions on a number line.
• Class 2 – Using a scale to measure weight and compare measurements to determine if items are heavier, lighter, or the same weight.
• Class 3 – Using integer arithmetic and number lines to investigate changes in temperature.

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Yesterday, I posted our intro to integer arithmetic using a problem set adapted from Project Z. Then a furious twitter session with Kent Haines and Michael Pershan led to today’s problem set.

We had a nice initial clarification about the parentheses in the third problem. The students were comfortable with the structure of the activity where they had to concentrate on representing their thinking (although we still got “I figured it out by using mental math,” to which I asked him for clarification so I could better understand his mental math strategy). Then they were off!

If we had used a traditional workshop model (mini-lesson, work, share, and reflect) for this lesson it could have started with this…

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Today we began our investigation of integers arithmetic by presenting the following problems pulled from Project Z.

The most important part for us as teachers today was the right hand column of this “worksheet.” Kristin Gray recently considered how a student’s representation of their thinking might differ based on the semantic difference of asking them to “show your work” or “show your thinking.” I decided to stay out of it and asked just to tell me how they figured it out. I’d love feedback on this prompt, as it is imperative that for students with cognitive processing challenges to represent their thinking with clarity, the directive must be as clear and concise as possible.

Anyway, back to integers. Kent Haines and Michael Pershan have pushed my thinking about integer instruction over the past couple months. I have always been an admirer of Cognitively Guided Instruction and be able to apply this instructional thinking to integers was too good to pass up.

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OR Students with Disabilities Investigating Social Justice Issues in Math Class

Graphs are important. Really important. Adults are often asked to make sense of graphs in order to make important decisions. The weather, graphs. At work, graphs. Presidential election, graphs. The interpretation and analysis of graphs is just as important as their creation. And as the election nears, our department identified graph and data analysis as a major point of focus for our students, especially when magazines and newspapers look like this…

Courtesy of The Guardian circa 2012 (http://www.theguardian.com/us)

So in an effort to help our student make sense of the graphs they see in their daily lives, we devised what follows. The guiding resources for our project were:

1. Standards for Mathematical Practice
2. Rethinking Mathematics: Teaching Social Justice by the Numbers
3. Making Sense of Graphs: Critical Factors Influencing Comprehension and Instructional Implications
4. NCTM Principles and Standards for School Mathematics
5. The GAISE Report

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Yesterday, I posted a planned I Notice/I Wonder activity that we adapted from an EngageNY worksheet.

Then the MathTwitterBlogoSphere got involved…

So, as requested, here’s how the activity went with the first group of students. The goal was to illuminate any misconceptions our students had in regards to measurement and learn from our mistakes. Growth mindset, anyone?

Our students noticed…

• There are black lines that are straight
• There are 5 boxes
• The brush is brown
• The paintbrush is made out of wood or plastic
• The paintbrush is 5 measuring long
• There are two lines on the side of the paintbrush

Our students wondered…

• How long is the paintbrush?
• Why are there lines on both sides?
• What are the squares for?
• How big is the paintbrush?
• What are the lines?
• What are the boxes?
• Why is the paintbrush not moving?

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Let me start by saying my students are never the reason for my bad days.  Never.  Usually my bad days come from the thought that I am not doing enough good for my students in a world that is in short supply of it.

However, instead of discussing the bad in the world, I wanted to share something good from today.

Today my students did their typical good, hard work using math to solve problems they might encounter in their daily lives. But this post is not about them, well, not directly.

It’s about my assistant teacher.  She is in her first year at our school and also finishing up her master’s in special education at Hunter College. I really enjoy talking with her about math and teaching. If you are a veteran teacher I highly recommend casually chatting about teaching with a new teacher. It really gets the juices going.

Anyway, so what was good about today you ask?

I’m getting there.

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Katharine Beals and Barry Garelick wrote an article in The Atlantic, in which the title of the article is pretty descriptive.

Dan Meyer was quick to post his response to this in a post titled, Understanding Math vs. Explaining Answers.  He concludes, “Let’s be inflexible in the goal but flexible about the many developmentally appropriate ways students can meet it.”

Twitter also responded to the article…

Which led me to ask whether my students, who typically struggle with expressive and receptive language based tasks, would benefit or be hindered by being asked to explain their thinking.

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